Continuous binomial distribution

Short description: Continuous probability distribution on the unit interval

Continuous binomial (cobin)

Parameters	$\theta \in \mathbb{ℝ}$ (natural parameter) $\lambda \in \{1,2,3,\dots \}$ (inverse dispersion)
Support	$x\in [0,1]$ if $\lambda =1$, $x\in (0,1)$ if $\lambda \ge 2$
PDF	$f(x;\theta ,\lambda )=h(x;\lambda )\exp (\lambda \theta x-\lambda B(\theta )),0\le x\le 1,$ with $B(\theta )=\{\begin{array}{ll}\log \left(\frac{e^{\theta }-1}{\theta }\right),& \theta \ne 0,\\ 0,& \theta =0,\end{array}$ and $h(x;\lambda )=\frac{\lambda }{(\lambda -1)!}\sum _{k=0}^{\lambda }(-1{)}^k(\genfrac{}{}{0ex}{}{\lambda }{k})\max (0,\lambda x-k{)}^{\lambda -1}.$
Mean	$B^{\prime }(\theta )=\{\begin{array}{ll}\frac{e^{\theta }}{e^{\theta }-1}-\frac{1}{\theta },& \theta \ne 0,\\ \frac{1}{2},& \theta =0,\end{array}$
Variance	$\frac{1}{\lambda }{B}^{″}(\theta )=\{\begin{array}{ll}\frac{1}{\lambda }(\frac{1}{{\theta }^2}-\frac{e^{\theta }}{(e^{\theta }-1{)}^2}),& \theta \ne 0,\\ \frac{1}{12\lambda },& \theta =0.\end{array}$

In probability theory and statistics, the continuous binomial distribution (also called the cobin distribution) is a family of continuous probability distributions on the unit interval that belongs to an exponential dispersion family. It was introduced as a response distribution for generalized linear models for continuous proportional data, proposed as an alternative to beta regression. The special case $\lambda =1$ coincides with the continuous Bernoulli distribution.

Definition

A random variable $X$ is said to follow a continuous binomial (cobin) distribution with natural parameter $\theta$ and inverse dispersion $\lambda \in \{1,2,\dots \}$, written $X\sim \mathrm{c}\mathrm{o}\mathrm{b}\mathrm{i}\mathrm{n}(\theta ,{\lambda }^{-1})$, if it has density on $[0,1]$ given by

$f(x;\theta ,\lambda )=h(x;\lambda )\exp (\lambda \theta x-\lambda B(\theta )),0\le x\le 1,$

where the log-partition function is

$B(\theta )=\{\begin{array}{ll}\log \left(\frac{e^{\theta }-1}{\theta }\right),& \theta \ne 0,\\ 0,& \theta =0,\end{array}$

and the base measure $h(x;\lambda )$ is

$h(x;\lambda )=\frac{\lambda }{(\lambda -1)!}\sum _{k=0}^{\lambda }(-1{)}^k(\genfrac{}{}{0ex}{}{\lambda }{k})\max (0,\lambda x-k{)}^{\lambda -1},$

with $h(x;1)=1$. The function $h(x;\lambda )/\lambda$ coincides with the probability density function of the Irwin–Hall distribution with parameter $n=\lambda$, evaluated at $\lambda x$.

When $\lambda$ is fixed, the cobin distribution belongs to a one-parameter natural exponential family in $\theta$.

Related distributions

• Continuous Bernoulli distribution: when $\lambda =1$, the cobin distribution reduces to the continuous Bernoulli distribution.

• Bates distribution: when $\theta =0$, the density reduces to $h(x;\lambda )$, corresponding to the distribution of the mean of $\lambda$ independent $\mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}(0,1)$ random variables (equivalently, a scaled Irwin–Hall distribution or Bates distribution).

• Uniform distribution: when $\lambda =1$ and $\theta =0$, the distribution reduces to the continuous uniform distribution on $[0,1]$.

• If $X_1,\dots ,X_{\lambda }$ are independent and identically distributed continuous Bernoulli random variables with common natural parameter $\theta$, then

$\overline{X}=\frac{1}{\lambda }\sum _{i=1}^{\lambda }{X}_i\sim \mathrm{c}\mathrm{o}\mathrm{b}\mathrm{i}\mathrm{n}(\theta ,{\lambda }^{-1}).$

Properties

Mean and variance

The mean and variance of $X\sim \mathrm{c}\mathrm{o}\mathrm{b}\mathrm{i}\mathrm{n}(\theta ,{\lambda }^{-1})$ can be expressed in terms of derivatives of $B(\theta )$:

• $\mathrm{E}(X)=B^{\prime }(\theta )=\frac{e^{\theta }}{e^{\theta }-1}-\frac{1}{\theta }$, for $\theta \ne 0$.
• $\mathrm{Var}(X)=\frac{1}{\lambda }{B}^{″}(\theta )=\frac{1}{\lambda }(\frac{1}{{\theta }^2}-\frac{e^{\theta }}{(e^{\theta }-1{)}^2})$, for $\theta \ne 0$.

If $\theta =0$, then $\mathrm{E}(X)=\frac{1}{2}$ and $\mathrm{Var}(X)=\frac{1}{12\lambda }$.

Sufficient statistic for the mean

If $X_1,\dots ,X_n$ are independent and identically distributed continuous binomial random variables with common natural parameter $\theta$ and fixed inverse dispersion parameter $\lambda$, then the sample mean

$\overline{X}=\frac{1}{n}\sum _{i=1}^n{X}_i$

is a sufficient statistic for $\theta$.

This is in contrast with the beta distribution: under a mean–precision parameterisation $X_i\sim \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(\mu \varphi ,(1-\mu )\varphi )$ with fixed $\varphi$, a sufficient statistic for the mean $\mu$ is

$\sum _{i=1}^n\log \left(\frac{X_i}{1-X_i}\right),$

not the sample mean $\overline{X}$.

Applications

The cobin distribution has been proposed as a response distribution for generalized linear models of continuous proportional data, as an alternative to beta regression, including extensions with random effects.